Almgren minimality of products of simplex-skeleton cones

In this paper, we prove that for every $m,n\geq 2$, the orthogonal product $S_m\times S_n\subset\R^m\times\R^n$ is an Almgren minimal cone of dimension $m+n-2$, where $S_k\subset\R^k$ is the cone over the $(k-2)$-skeleton of a regular $k$-simplex. Lawlor and Morgan \cite{LM94} proved the minimality